Tiles and TessellationsDuke TIP Academic Adventures

Game of SET

A set consists of 3 cards in which each of the cards features, looked at one-by-one, are the same on each card, or different on each card.

Game of SET

Game of SET● At your table deal 12 cards face up.

● Players do not take turns, but pick up SETs as they see them. If everyone agrees there are no SETs, deal 3 more cards on the table.

● Replace the cards from the top of the deck when a SET is removed.

● The person with the most SETs wins!

Third Person IntroductionsIntroduce the person sitting next you by saying the following.● Name

● Hometown

● Favorite Food

Human Machine

Rules and Expectations● Be Respectful● Participate● Don’t Talk When Someone Else Is● Don’t Shout Out Answers● Thinking > Answers● Have Fun

Finishing EarlyEveryone works at a different pace. If you you happen to finish something early, please raise your hand and let me know. You may get to be my assistant!

Warm UpDraw the line(s) of symmetry on each figure.

Warm Up

Tiles and Tessellations

Tiles and TessellationsA tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

Create a TessellationCreate your own tessellations using the shapes provided. You may use as many different shapes as you like.

Create a Tessellation● How did you create your tessellations?

● Could you come up with a way to create a tessellation, if given a few shapes?

PolygonsA polygon is a closed geometric figure.

Polygon Not a Polygon

Convex and Concave Polygons

Polygon Angle SumWhat is the sum of the angle measures of a triangle?

Polygon Angle Sum1. Draw a triangle on a sheet of paper using a ruler.

2. Carefully cut out the triangle with the scissors provided.

3. Rip off two of the corners.

4. Place them next to the remaining corner so that they all share a vertex.

5. What do you notice?

Polygon Angle SumThe sum of the angle measures of a triangle is 180 degrees!

Polygon Angle SumWhat is the sum of the angle measures of a quadrilateral (rectangle, square, rhombus, parallelogram, trapezoid, kite, etc.)?

Polygon Angle SumHint: Separate the quadrilateral into two triangles.

Polygon Angle SumThe sum of the angle measures of a quadrilateral is 360 degrees!

Polygon Angle Sum● What is the sum of the angle measures of pentagon (5 sided

polygon)? Hexagon (6 sided polygon)?

● Create a table and record these values.

● Can you predict the sum of the angle measures a polygon with any number of sides?

Polygon Angle Sum

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3 180

Quadrilateral 4 360

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3 180

Quadrilateral 4 360

Pentagon 5 540

Hexagon 6 720

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3 180

Quadrilateral 4 360

Pentagon 5 540

Hexagon 6 720

Heptagon 7 900

Octagon 8 1080

Nonagon 9 1260

Decagon 10 1440

n-gon n

100-gon 100

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3 180

Quadrilateral 4 360

Pentagon 5 540

Hexagon 6 720

Heptagon 7 900

Octagon 8 1080

Nonagon 9 1260

Decagon 10 1440

n-gon n 180(n-2)

100-gon 100

Polygon Angle SumPolygon Number of Sides Sum of Angle Measures

Triangle 3 180

Quadrilateral 4 360

Pentagon 5 540

Hexagon 6 720

Heptagon 7 900

Octagon 8 1080

Nonagon 9 1260

Decagon 10 1440

n-gon n 180(n-2)

100-gon 100 17640

Polygon Angle SumWhy does the formula S = 180(n-2) make sense?

Polygon Angle SumHint: How many triangles can you create inside of a polygon with n sides?

Find the Measure of the Missing Angle

Find the Measure of the Missing Angle

39 + 55 + x = 18094 + x = 180

x = 86

75 + 100 + 110 + x = 360285 + x = 360

x = 75

68 + 121 + 103 + 85 + x = 540377 + x = 540

x = 163

Challenge ProblemFind the value of x, then find the measures of the missing angles.

Challenge Problem5x + 3 + 88 + 10x + 7 + 127 = 360

15x + 225 = 36015x = 135

x = 9

The two missing angle measures are 5(9) + 3 = 48 and 10(9) + 7 = 97.

Polygon Angle SumIs there a polygon with an angle sum of 1980 degrees? If so, how many sides does it have? If not, explain why.

Polygon Angle SumYes, it has 13 sides!

180(n-2) = 1980180n - 360 = 1980

180n = 2340n = 13

Polygon Angle SumIs there a polygon with an angle sum of 2960 degrees? If so, how many sides does it have? If not, explain why.

Polygon Angle SumNo!

If this were true we would have180(n-2) = 2960

180n - 360 = 2960180n = 3320

n ≈ 18.4Since n is the number of sides, n must be a whole number.

Create a Tessellation

Regular PolygonsA regular polygon is a polygon with congruent sides and angles. This means all of the sides have the same length and all of the angles have the same measure.

Regular Polygons

Regular PolygonsNow that we know the sum of the angle measures of any polygon, let’s find the measure of each angle in a regular polygon. We will call this angle the corner angle.

Regular PolygonsWhat is the measure of each angle in an equilateral triangle?

Regular Polygons

Regular Polygons● What about a square? Regular pentagon? Regular Hexagon?

● Create a table and record these values.

● Can you come up with a formula for the corner angle of a regular n-gon?

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3

Square 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3 60

Square 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3 60

Square 4 90

Pentagon 5 108

Hexagon 6 120

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

n-gon n

100-gon 100

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3 60

Square 4 90

Pentagon 5 108

Hexagon 6 120

Heptagon 7 128.6

Octagon 8 135

Nonagon 9 140

Decagon 10 144

n-gon n

100-gon 100

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3 60

Square 4 90

Pentagon 5 108

Hexagon 6 120

Heptagon 7 128.6

Octagon 8 135

Nonagon 9 140

Decagon 10 144

n-gon n 180(n-2)/n

100-gon 100

Regular Polygon Corner AngleRegular Polygon Number of Sides Measure of Corner Angle

Equilateral Triangle 3 60

Square 4 90

Pentagon 5 108

Hexagon 6 120

Heptagon 7 128.6

Octagon 8 135

Nonagon 9 140

Decagon 10 144

n-gon n 180(n-2)/n

100-gon 100 176.4

Regular Polygon Corner AngleHow big can a corner angle of a regular polygon get? Why?

Regular Polygon Corner Angle

Regular Polygon Corner AngleA corner angle can not exceed 180 degrees, since this would be a straight line.

Regular Polygon Corner AngleIf a polygon has a corner angle measure of 150, can it be a regular polygon? If so, how many sides does it have? If not, explain why.

Regular Polygon Corner AngleYes, it has 12 sides!

180(n-2)/n = 150180(n-2) = 150n180n-360 = 150n

30n - 360 = 030n = 360

n = 12

Regular Polygon Corner AngleIf a polygon has a corner angle measure of 100, can it be a regular polygon? If so, how many sides does it have? If not, explain why.

Regular Polygon Corner AngleNo!

180(n-2)/n = 100180(n-2) = 100n180n-360 = 100n

80n - 360 = 080n = 360

n = 4.5Since n is the number of sides, n must be a whole number.

Lunch

Regular TessellationsRegular Polygon Tile (Yes or No) Explanation Picture (If Possible)

Equilateral Triangle Yes

Square Yes

Pentagon No

Hexagon Yes

Heptagon No

Octagon No

Regular TessellationsCan we tile the plane with any regular polygon?

Regular Tessellations● In the table you just created, there are only three regular

polygons that can be used to tile the plane.

● Determine which three can be used to tile the plane.

● Explain why those three can be used to tile the plane and why the other five cannot.

Regular Tessellations

Regular TessellationsCorners of the tiles need to fit together around a point, which means the corner angle of the regular polygon must evenly divide 360.

Regular TessellationsRegular Polygon Tile (Yes or No) Explanation Picture (If Possible)

Equilateral Triangle Yes 360/60 = 6

Square Yes 360/90 = 4

Pentagon No 360/108 ≈ 3.33

Hexagon Yes 360/120 = 3

Heptagon No 360/128.6 ≈ 2.8

Octagon No 360/135 ≈ 2.66

Regular TessellationsCan we tile the plane with any polygon with more than 6 sides? Why or why not?

Regular Tessellations● No!

● A regular polygon with more than 6 sides has a corner angle larger than 120 degrees and smaller than 180 degrees.

● Is there any number between 120 and 180 that divides into 360 evenly?

Regular Tessellations

Equilateral Triangle Square Regular Hexagon

Semi-Regular TessellationsA semi-regular tessellation is a tiling of the plane made up of two or more regular polygons such that the same polygons are in the same order surrounding each vertex.

Semi-Regular Tessellations● There are 5 other semi-regular tessellations.

● Create them using the shapes provided and sketch what each one looks like.

● Explain why there are no other semi-regular tessellations.

Semi-Regular Tessellations

Semi-Regular Tessellations

How can we move shapes?

How can we move shapes?● Flips

● Turns

● Slides

Reflection (Flip)● A figure can be reflected (flipped) across a line

of symmetry (mirror line).

● Can you think of any real lifeexamples?

Reflection (Flip)

Rotation (Turn)● A figure can be rotated (turned) about a point of

rotation (center point).

● Can you you think of any real lifeexamples?

Rotation (Turn)

Translation (Slide)● A figure can be translated (slid) along a vector.

● Can you you think of any real lifeexamples?

Translation (Slide)

Transformations● Reflections

● Rotations

● Translations

Symmetries● A symmetry is a transformation that moves a

figure onto itself.

● A figure can have either reflectional or rotational symmetry.

SymmetriesHow many symmetries does an equilateral triangle have?

Symmetries

SymmetriesHow many symmetries does a square have?

Symmetries

SymmetriesHow many symmetries does regular pentagon have?

Symmetries

SymmetriesCan you predict the number of rotations that an n sided regular polygon has?

SymmetriesAn n sided regular polygon has n reflectional symmetries and n rotational symmetries, a total of 2n symmetries.

How are transformations and symmetries related to tessellations?

Map ColoringWhat is the minimum number of colors required to color any “map” so that adjacent edges are not the same color?

Map ColoringTry to color these using the least amount of colors, where adjacent edges must be different colors.

Map Coloring

Four Color TheoremIt turns out that any “map” can be colored with four colors or less.

Tessellation ColoringColor the tessellation you created earlier so that no adjacent sides are the same color.

Evaluations and Certificates1. Fill out the evaluation for Duke TIP.2. Write down what you liked and didn’t like about

the class on a blank sheet of paper.3. Receive your certificate.4. Celebrate!